Integer frequency offset estimation in wireless systems

ABSTRACT

Systems and techniques relating to wireless communications are described. A described technique includes receiving a plurality of symbols, observing a plurality of data samples in adjacent symbols, and calculating an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols. Calculating the estimate can include calculating sum values corresponding to respective symbol indices. Each of the sum values can be based on a summation of max values that correspond to respective data subcarrier indices, the max values being based on a maximum of an absolute value of a real component of a base value and an absolute value of an imaginary component of the base value, where the base value is based on at least one of the data samples.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation (and claims the benefit of priority under 35 USC 120) of U.S. application Ser. No. 11/031,614, entitled “INTEGER FREQUENCY OFFSET ESTIMATION BASED ON THE MAXIMUM LIKELIHOOD PRINCIPAL IN OFDM SYSTEMS,” filed Jan. 7, 2005, and issued on Mar. 17, 2009, as U.S. Pat. No. 7,505,523, which claims priority to U.S. Provisional Application Ser. No. 60/600,877, entitled “INTEGER FREQUENCY OFFSET ESTIMATION BASED ON THE MAXIMUM LIKELIHOOD PRINCIPAL IN OFDM SYSTEMS,” filed on Aug. 12, 2004.

BACKGROUND

Wireless systems may use an Orthogonal Frequency Division Multiplexing (OFDM) transmission scheme. In an OFDM system, a data stream is split into multiple substreams, each of which is sent over a subcarrier frequency. Because data is carried over multiple carrier frequencies, OFDM systems are referred to as “multicarrier” systems as opposed to single carrier systems in which data is transmitted on one carrier frequency.

An advantage of OFDM systems over single carrier systems is their ability to efficiently transmit data over frequency selective channels by employing a fast Fourier transform (FFT) algorithm instead of the complex equalizers typically used by single carrier receivers. This feature enables OFDM receivers to use a relatively simple channel equalization method, which is essentially a one-tap multiplier for each tone.

Despite these advantages, OFDM systems may be more sensitive to carrier frequency offset, which is the difference between the carrier frequency of the received signal and the local oscillator's frequency. The carrier frequency offset may be caused by Doppler shift and oscillator instabilities, and can be many times larger than the subcarrier bandwidth.

SUMMARY

The estimator may include a receiver to receive symbols, each symbol including a plurality of data samples, a framer to observe a plurality of data samples in adjacent symbols, and a calculator to calculate an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols.

The estimator may calculate the estimate by solving the equation:

${\hat{l} = {\underset{n}{argmax}\left\{ {T(n)} \right\}}},\mspace{14mu}{{{where}\mspace{14mu}{T(n)}} = {\sum\limits_{k \in S_{2}}^{\;}\;{\sigma_{z}^{2}{\log\left( {{\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)} + {\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)}} \right)}}}}$

where n is a symbol index, k is a subcarrier index, S₂ is a set of indices for data subcarriers, σ_(z) ² is a variance, α is a cyclic prefix width ratio, and V is an observation vector. Alternatively, the estimator may calculate the estimate by solving the equation:

$\hat{l} = {\underset{n}{argmax}\left\{ {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}} \right\}}$

where n is a symbol index, k is a subcarrier index, S₂ is a set of indices for data subcarriers, α is a cyclic prefix width ratio, and V is an observation vector.

In another embodiment, the estimator may calculate the estimate of the integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data subcarriers and the pilot subcarriers between OFDM symbols. The estimator may calculate the estimate by solving the equation:

${\hat{l} = {\underset{n}{argmax}\left\{ {T(n)} \right\}}},\mspace{14mu}{where}$ $\begin{matrix} {{T(n)} = {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} +}} \\ {\sum\limits_{k \in S_{2}}^{\;}{\sigma_{z}^{2}{\log\left( {{\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)} +} \right.}}} \\ \left. {\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)} \right) \end{matrix}$

where n is a symbol index, k is a subcarrier index, S₁ is a set of indices for pilot subcarriers, S₂ is a set of indices for data subcarriers, σ_(z) ² is a variance, α is a cyclic prefix width ratio, V is an observation vector, and A is a sequence known to the receiver. Alternatively, the estimator may calculate the estimate by solving the equation:

$\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}}} \right\}}$

where n is a symbol index, k is a subcarrier index, S₁ is a set of indices for pilot subcarriers, S₂ is a set of indices for data subcarriers, α is a cyclic prefix width ratio, V is an observation vector, and A is a sequence known to the receiver.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a wireless system according to an embodiment.

FIG. 2 is a flowchart describing an integer carrier offset estimation operation according to an embodiment.

FIG. 3 is a plot showing the performance of blind estimators for an additive white Gaussian noise (AWGN) channel.

FIG. 4 is a plot showing the performance of pilot-aided estimators for an AWGN channel.

FIG. 5 is a plot showing the performance of blind estimators for multipath fading channels.

FIG. 6 is a plot showing the performance of pilot-aided estimators for multipath fading channels.

DETAILED DESCRIPTION

FIG. 1 shows a wireless communication system 100 according to an embodiment. The wireless communication system includes a transmitter 102 and a receiver 104 that communicate over a wireless channel 106. The transmitter 102 and receiver 104 may be implemented in two different transceivers, each transceiver including both a transmitter section and a receiver section.

The wireless communication system 100 may be implemented in a wireless local Area Network (WLAN) that complies with the IEEE 802.11 standards (including IEEE 802.11a, 802.11g, and 802.11n). The IEEE 802.11 standards describe OFDM systems and the protocols used by such systems.

In an OFDM system, a data stream is split into multiple substreams, each of which is sent over a different subcarrier frequency (also referred to as a “tone”). For example, in IEEE 802.11a systems, OFDM symbols include 64 tones (with 48 active data tones) indexed as {−32, −31, . . . , −1, 0, 1, . . . , 30, 31}, where 0 is the DC tone index. The DC tone is not used to transmit information.

The transmitter 102 may generate OFDM symbols for transmission to the receiver 104. An inverse fast Fourier transform (IFFT) module 108 may generate the m-th OFDM symbol x_(m)[n] by performing an N-point IFFT on the information symbols X_(m)[k] for k=0, 1 . . . , N−1. A cyclic prefix may be added to the body of the OFDM symbol to avoid interference (ISI) and preserve orthogonality between subcarriers. The cyclic prefix may include copies of the last N_(g) samples of the N time-domain samples. The cyclic prefix is appended as a preamble to the IFFT of X_(m)[k] to form the complete OFDM symbol with N_(t)=N_(g)+N samples.

The OFDM symbols are converted to a single data stream by a parallel-to-serial (P/S) converter 110 and concatenated serially. The discrete symbols are converted to analog signals by a digital-to-analog converter (DAC) 112 and lowpass filtered for radio frequency (RF) upconversion by an RF module 114.

The OFDM symbols are transmitted through the wireless channel h_(m)[n] 106 over a carrier to the receiver 104, which performs the inverse process of the transmitter 102. The carrier may be corrupted by Gaussian noise z_(m)[n] in the channel, which is assumed to be block stationary, i.e., time-invariant during each OFDM symbol.

At the receiver 104, the passband signal is downconverted and filtered by an RF module 120 and converted to a digital data stream by an analog-to-digital converter (ADC) 122.

The RF module includes a local oscillator 123. When the local oscillator frequency f_(l) is not matched to the carrier frequency f_(c) of the received signal, a carrier frequency offset Δf=f_(c)−f_(l) will appear. In addition, there may also be a phase offset θ₀. The received symbol y_(m)[n] can then be represented as y _(m) [n]=e ^(f[2)πΔf(n+m(N+N ^(g) ^())T+θ) ⁰ ^(])(h _(m) [n]*x _(m) [n])+ z _(m) [n],  (1)

where T is the sampling period, and z_(m)[n] is a zero-mean complex-Gaussian random variable with variance σ_(z) ².

The frequency offset Δf can be represented with respect to the subcarrier bandwidth 1/NT by defining the relative frequency offset ε as

$\begin{matrix} {{ɛ\overset{\Delta}{=}{\frac{\Delta\; f}{1/{NT}} = {\Delta\;{fNT}}}},} & (2) \end{matrix}$

With the above definition of the relative frequency offset ε, the received symbol y_(m)[n] can be expressed as

$\begin{matrix} {{{y_{m}\lbrack n\rbrack} = {{{\mathbb{e}}^{j\frac{2{\pi ɛ}\; n}{N}}{\mathbb{e}}^{{j2\pi ɛ}\;{m{({1 + \alpha})}}}{{\mathbb{e}}^{{j\theta}_{0}}\left( {{h_{m}\lbrack n\rbrack}*{x_{m}\lbrack n\rbrack}} \right)}} + {z_{m}\lbrack n\rbrack}}},\mspace{14mu}{{{where}\mspace{14mu}\alpha} = {\frac{N_{g}}{N}.}}} & (3) \end{matrix}$

The relative frequency offset ε can be divided into an integer part l and a fractional part ε such that ε=l+ ε,  (4)

where l is an integer and −½≦ ε≦½.

When the fractional frequency offset is equal to zero, the discrete Fourier transform (DFT) of y_(m)[n] can be expressed as Y _(m) [k]=e ^(j(2πlmα+θ) ⁰ ⁾ H _(m) [k−l]+X _(m) [k−l]+Z _(m) [k],  (5)

where H_(m)[n] and Z_(m)[n] are the DFTs of h_(m)[n] and z_(m)[n], respectively. In Equation (5) it is assumed that H_(m)[n] and X_(m)[n] are periodic with period N to simplify the notation. Similarly, Y_(m)[n] is also be assumed to be periodic.

As can be seen in Equation (5), the integer frequency offset l causes a cyclic shift and a phase change of the received signal. In an embodiment, an estimator 124 may be used to estimate the integer frequency offset l. The fractional part ε of the relative frequency offset may be calculated separately. The estimated integer frequency offset {circumflex over (l)} may be fed back to the downconverter 120 to correct any cyclic shift in the subcarriers. The data stream is then converted into parallel substreams by a serial-to-parallel (S/P) converter 128 and transformed into N tones by an FFT module 130.

In some implementations, the estimator 124 can include a receiver 150 to receive symbols, each symbol can include a plurality of data samples, a framer 152 to observe a plurality of data samples in adjacent symbols, and a calculator 154. The calculator 155 can calculate an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols.

In an embodiment, the estimator 124 may perform either pilot-aided estimation or blind estimation of the integer frequency offset l, i.e., with or without the aid of pilot subcarriers. In a pilot-aided OFDM system, the subcarriers include pilot subcarriers, data subcarriers, and unused subcarriers. In a blind OFDM system, the subcarriers do not include pilot subcarriers and the estimators utilize redundant information in the data subcarriers to estimate the carrier frequency offset.

In the following equations, S₁ and S₂ represent the set of indices for pilot subcarriers and data subcarriers, respectively. The number of elements in S₁ and S₂ are N₁ and N₂, respectively. Depending on the particular OFDM system and OFDM symbol, N₁ or N₂ may be zero.

The transmit symbols satisfy the following relationship:

$\begin{matrix} {{{X_{m}^{*}\lbrack k\rbrack}{X_{m}\lbrack k\rbrack}} = \left\{ \begin{matrix} {A_{m}\lbrack k\rbrack} & {for} & {k \in S_{1}} \\ {B_{m}\lbrack k\rbrack} & {for} & {{k \in S_{2}},} \end{matrix} \right.} & (6) \end{matrix}$

where A_(m)[k] is a sequence known to the receiver, B_(m)[k] is a random sequence unknown to the receiver, and A_(m)[k] and B_(m)[k] are assumed to have power of 1.

In an embodiment, the ML estimator for the integer frequency offset l is derived for the additive white Gaussian noise (AWGN) channel. For the AWGN channel, the received signal can represented as follows: Y _(m) [k]=e ^(j(2πlmα+θ) ⁰ ⁾ X _(m) [k−l]+Z _(m) [k].  (7)

Since the phase θ₀ is unknown to the receiver, Y_(m)[k] is multiplied by Y*_(m-1)[k] to remove the phase term θ₀ from the desired signal X_(m)[k−l] and X_(m-l)[k−l]: V _(m) [k]=Y* _(m-1) [k]Y _(m) [k]=e ^(j2πlmα) X* _(m-1) [k−l]·X _(m) [k−l]+Z′ _(m) [k]  (8)

where the noise Z′_(m)[k] is Z′ _(m) [k]=e ^(−j(2πl(m-1)α+θ) ⁰ ⁾ X* _(m-1) [k−l]+Z _(m) [k]+e ^(j(2πlmα+θ) ⁰ ⁾ X _(m) [k−l]Z* _(m-1) [k]+Z* _(m-l) [k]Z _(m) [k]  (9)

For the derivation of a closed-form ML estimator, it may be assumed that Z*_(m-l)[k]Z_(m)[k] is negligible compared to the other terms in Equation (9). This assumption may be valid when the SNR is high. Under this assumption, Z*_(m)[k] will approximately follow a Gaussian distribution.

The estimator 124 performs an ML estimation of the integer frequency offset using samples in an observation window. Let V be an observation vector comprised of the observations V_(m)[0], V_(m)[1], . . . , V_(m)[N−1], i.e., V=[V_(m)[0]V_(m)[1] . . . V_(m)[N−1]]. The ML estimate of l given the observation V is the integer n that maximizes the following statistic: T ₁(n)=f(V|l=n),  (10)

where f is the conditional probability density function (pdf) of V given l=n. Since the observation vector V depends not only on the integer frequency offset l but also on the values of the data symbols B_(m)[k] for k_(l)εS₂, the conditional pdf f(V|B=b,l=n) can be rewritten as follows:

$\begin{matrix} {{{T_{1}(n)} = {\sum\limits_{all\_ b}^{\;}\;{{f\left( {{{V❘B} = b},{l = n}} \right)}P\left\{ {B = b} \right\}}}},} & (11) \end{matrix}$

where B=[B[k₁], B_(m)[k₂], . . . , B_(m)[k_(N) ₂ ]] for k_(i)εS₂ and b=[b_(m)[K₁], b_(m)[k₂], . . . , b_(m)[k_(N) ₂ ]] represents the actual value assumed by the random vector B. It may be assumed that the data subcarriers are modulated using the quadrature phase shift key (QPSK) technique. The ML estimator for the other constellations can be derived in a similar manner.

Since P{B=b}=2^(−2N) ² for QPSK, the following T₂(n) can be used instead of T₁(n) for the ML estimation:

$\begin{matrix} {{T_{2}(n)} = {\sum\limits_{all\_ b}^{\;}\;{{f\left( {{{V❘B} = b},{l = n}} \right)}.}}} & (12) \end{matrix}$

Since the received signal in each subcarrier is independent of one another, the conditional pdf f(V|B=b,l=n) becomes

$\begin{matrix} \begin{matrix} {{f\left( {{{V❘B} = b},{l = n}} \right)} = {\prod\limits_{k \in S_{1}}^{\;}\;{{f_{z^{\prime}}\left( {{V\left\lbrack {k + n} \right\rbrack} - {{\mathbb{e}}^{{j2\pi}\; n\;\alpha}{A\lbrack k\rbrack}}} \right)} \cdot}}} \\ {\prod\limits_{k \in S_{2}}^{\;}\;{{f_{z^{\prime}}\left( {{V\left\lbrack {k + n} \right\rbrack} - {{\mathbb{e}}^{{j2\pi}\; n\;\alpha}{b\lbrack k\rbrack}}} \right)}\mspace{14mu}{where}}} \end{matrix} & (13) \\ {{f_{z^{\prime}}(x)} = {\frac{1}{2{\pi\sigma}_{z}^{2}}{{\mathbb{e}}^{\frac{- {z}^{2}}{2\sigma_{z}^{2}}}.}}} & (14) \end{matrix}$

From Equation (14), it can be shown that

$\begin{matrix} {{{f_{z^{\prime}}\left( {{V\left\lbrack {k + n} \right\rbrack} - {{\mathbb{e}}^{{j2\pi}\; n\;\alpha}{A\lbrack k\rbrack}}} \right)} = {\frac{1}{2{\pi\sigma}_{z}^{2}}{\mathbb{e}}^{- \frac{{{V{\lbrack{k + n}\rbrack}}}^{2} + {{A{\lbrack k\rbrack}}}^{2}}{2\sigma_{z}^{2}}}{\mathbb{e}}^{\frac{\{{{V{\lbrack{k + n}\rbrack}}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}{\lbrack k\rbrack}}}\}}{\sigma_{z}^{2}}}}},} & (15) \end{matrix}$

where

{x} denotes the real part of x. Using the expression in Equation (13) and removing all factors that are independent of n, it can be shown that the ML estimator maximizes T₃(n):

$\begin{matrix} {{T_{3}(n)} = {\sum\limits_{all\_ b}^{\;}\;{\left( {\prod\limits_{k \in S_{1}}^{\;}\;{\mathbb{e}}^{\frac{\{{{V{\lbrack{k + n}\rbrack}}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}{\lbrack k\rbrack}}}\}}{\sigma_{z}^{2}}}} \right){\left( {\prod\limits_{k \in S_{2}}^{\;}\;{\mathbb{e}}^{\frac{\{{{V{\lbrack{k + n}\rbrack}}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{b^{*}{\lbrack k\rbrack}}}\}}{\sigma_{z}^{2}}}} \right).}}}} & (16) \end{matrix}$

Since each b[k_(i)] for k_(i)εS₂ can take only the values ±1 and ±j and the b[k_(i)] values are independent of one another, T₃(n) can be rewritten as

$\begin{matrix} {{T_{3}(n)} = {\prod\limits_{k \in S_{1}}^{\;}{{\mathbb{e}}^{\frac{\{{{V{\lbrack{k + n}\rbrack}}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}{\lbrack k\rbrack}}}\}}{\sigma_{z}^{2}}} \cdot {\prod\limits_{k \in S_{2}}^{\;}{2\left( {{\cosh\left( \frac{\left\{ {{V\left\lbrack {{k + n}❘} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}{\sigma_{z}^{2}} \right)} + {\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}{\sigma_{z}^{2}} \right)}} \right)}}}}} & (17) \end{matrix}$

where

{x} denotes the imaginary part of x. By taking the logarithm of T₃(n), removing the constant terms, and multiplying by α_(z) ², it can be shown that the ML estimate of the integer frequency offset given the observation V is

$\begin{matrix} {{\hat{l} = {\underset{n}{argmax}\left\{ {T(n)} \right\}}},\mspace{14mu}{where}} & (18) \\ {{{T(n)} = {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}{\sigma_{z}^{2}{\log\left( {{\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)} + {\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)}} \right)}}}}}{{for}\mspace{14mu} a\mspace{14mu}{pilot}\text{-}{aided}\mspace{14mu}{estimator}\mspace{14mu}\left( {P\; A\; E} \right)\mspace{14mu}{and}}} & (19) \\ {{T(n)} = {\sum\limits_{k \in S_{2}}^{\;}{\sigma_{z}^{2}{\log\left( {{\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)} + {\cosh\left( \frac{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}{\sigma_{z}^{2}} \right)}} \right)}}}} & (20) \end{matrix}$

for a blind estimator (BE).

FIG. 2 is a flowchart describing an ML estimation of the integer frequency offset according to an embodiment. The receiver 104 receives OFDM symbols from the channel (block 202). The cyclic prefix is removed and FFT performed on the symbols. The estimator 124 then observes samples in the observation window V (block 204). The estimator 124 uses the fact that the integer carrier frequency offset causes a phase shift over two OFDM symbols and a cyclic shift of the subcarriers to calculate the estimated integer frequency offset (block 206). The cyclic shift of the subcarriers is reflected on the shift of the index of V[k] in the calculation of T(n), whereas the phase shift over two OFDM symbols is exploited as follows. In the first summation in Equation (19), each term measures the component of the observation V[k+n] for k_(i)εS₁ in the direction of e^(12πnα)A[k]. In the absence of noise, each term should be equal to 1 when the estimate is equal to the actual integer frequency offset l. However, it will have a value less than 1 when the estimate n is different from l. In the second summation, each term measures the magnitude of the observation V[k+n] for k_(i)εS₂ in the direction of ±e^(j2πnα) and

$\pm {\mathbb{e}}^{j{({{2\pi\; n\;\alpha} + \frac{\pi}{2}})}}$ and takes the average with a function comprised of log and cos h. In the absence of noise, each term should be equal to

$\sigma_{z}^{2}{\log\left( {\cosh\left( \frac{1}{\sigma_{z}^{2}} \right)} \right)}$ when (n−l)α is an integer multiple of

$\frac{1}{4}.$ It will have a smaller value otherwise.

The estimator 124 can use the ML estimator of Equations (18), (19), and (20) to perform “exact” ML estimation of the integer frequency offset. The estimator may also perform an approximate ML estimation of the integer frequency offset by simplifying the statistic T(n) for high SNR. Equation (19) can be simplified by noting that

$\begin{matrix} {{{{\cosh(x)} + {\cosh(y)}} = {{\frac{1}{2}\left( {{\mathbb{e}}^{x} + {\mathbb{e}}^{- x} + {\mathbb{e}}^{y} + {\mathbb{e}}^{- y}} \right)} \approx {\frac{1}{2}{\mathbb{e}}^{\max{\{{{x},{y}}\}}}}}},} & (21) \end{matrix}$

for |x|>>|y|>>0 or |y|>>|x|>>0. With this approximation, the pilot-aided estimator (PAE) can be represented as:

$\begin{matrix} {\hat{l} = {\underset{n}{argmax}{\left\{ {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}}} \right\}.}}} & (22) \end{matrix}$

For a blind estimator (BE), Equation (21) becomes

$\begin{matrix} {\hat{l} = {\underset{n}{argmax}{\left\{ {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}} \right\}.}}} & (23) \end{matrix}$

The performance of the exact ML estimators of Equations (18), (19), and (20) and the approximate ML estimators of Equations (22) and (23) are compared with the performance of known PAEs and BEs in FIGS. 3-6. The known estimators are described in T. M. Schmidl and D. C. Cox, “Blind synchronisation for OFDM,” Electronics Letters, vol. 33, pp. 113-114, January 1997, T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613-1621, December 1997, M. Morelli, A. N. D'Andrea, and U. Mengali, “Frequency ambiguity resolution in OFDM systems,” IEEE Commun. Lett., vol. 4, pp. 134-136, April 2000.

Schmidl's PAE is given by

$\begin{matrix} {{\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{2}}^{\;}{{V\left\lbrack {k + n} \right\rbrack}{A^{*}\lbrack k\rbrack}}}}^{2} \right\}}},} & (24) \end{matrix}$

and Schmidl's BE is given by

$\begin{matrix} {\hat{l} = {\underset{n}{argmax}{\left\{ {\sum\limits_{k \in S_{2}}^{\;}\frac{\left( {{V\lbrack k\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right)^{4}}{{{V\lbrack k\rbrack}}^{3}}} \right\}.}}} & (25) \end{matrix}$

Morelli's PAE is given by

$\begin{matrix} {{\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{1}}^{\;}{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\frac{1}{2}{\sum\limits_{k \in {S_{1}\bigcup S_{2}}}^{\;}{\sum\limits_{i = {m - 1}}^{m}\;{{Y_{i}\left\lbrack {k + n} \right\rbrack}}^{2}}}}} \right\}}},} & (26) \end{matrix}$

(26)

and Morelli's BE is given by

$\begin{matrix} {\hat{l} = {\underset{n}{argmax}{\left\{ {\sum\limits_{k \in S_{2}}^{\;}{\sum\limits_{i = {m - 1}}^{m\;}{{Y_{i}\left\lbrack {k + n} \right\rbrack}}^{2}}} \right\}.}}} & (26) \end{matrix}$

The following parameters were chosen for the simulations. The symbol in each subcarrier is taken from a QPSK constellation. The number of data samples N and the number of cyclic prefix samples N_(g) in one OFDM symbol are 256 and 14, respectively, resulting in cyclic prefix width ratio α of

$\frac{N_{g}}{N} = {\frac{7}{128}.}$ The maximum magnitude of the integer frequency offset is chosen as 4. The number of pilot subcarriers N₁ is fixed at 7, whereas the number of data subcarriers N₂ is chosen to be either 249 or 241. Thus, the number of used subcarriers N_(u)=N₁+N₂ is either 256 or 248.

FIG. 3 illustrates the probability of the failure of the BEs for the AWGN channel. Schmidl's, Morelli's, exact ML, and approximate ML BEs are compared. For N_(u)=256, Schmidl's BE performs much better than Morelli's BE. This can be explained as follows. When all the subcarriers are used, a rotation of the subcarriers cannot be detected by measuring the energy of used subcarriers, i.e., the summation in Morelli's BE is constant regardless of n. The approximate ML BE, which uses both the phase shift property and the subcarrier rotation property, performs as well as Schmidl's BE. As can be seen from the plot, there is little loss in performance when the approximate ML BE is used instead of the exact ML BE. For N_(u)=248, the performance of Schmidl's BE almost remains the same as for N_(u)=256, whereas the performance of Morelli's BE improves significantly because Morelli's BE uses the property that the integer frequency offset causes a cyclic shift of subcarriers. However, Morelli's BE is outperformed by the exact and approximate ML BEs for high SNR. As expected from N_(u)=248, the performance of the approximate ML BE is almost as good as that of the exact ML BE for N_(u)=256.

FIG. 4 illustrates the probability of the failure of the PAEs for the AWGN channel. Schmidl's PAE performance does not depend of the number of used subcarriers N_(u) but only on the number of the pilot subcarriers N₁. On the other hand, the performance of Morelli's PAE and the ML PAEs depend not only on N₁ but also N_(u). Since Morelli's PAE and the ML PAEs exploit the fact that the frequency offset causes a cyclic shift of subcarriers, the performance of Morelli's PAE and the ML PAEs is better for N_(u)=248 than for N_(u)=256. It can also be seen from the from the plot that Schmidl's PAE performs worse than both Morelli's PAE and the approximate ML PAE, and Morelli's PAE is outperformed by the approximate ML PAE for high SNR. Unlike the case for the blind estimators, the performance difference between the exact ML PAE and the approximate ML PAE is not negligible. However, the performance difference decreases as the SNR increases, as expected.

FIGS. 5 and 6 show the probability of failure of the BEs and PAEs, respectively, for multipath fading channels. The multipath channel used in the simulation includes fifteen paths, each of which vary independently of each other with magnitude following the Rayleigh distribution and phase following the uniform distribution. The fifteen paths have an exponential power delay profile, and the root-mean-square delay spread of the multipath channel is two samples.

The simulation results for the multipath fading channel are similar to the results for the AWGN channel except that Morelli's BE performs worse than Schmidl's BE for N_(u)=248 and SNR larger than 6 dB. These results indicate that the exact and approximate ML estimators developed for the AWGN channel perform better than existing estimators even for the multipath channel, and especially for high SNR.

A number of embodiments have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, the integer frequency offset estimator can be implemented in other OFDM systems include, for example, fixed wireless access and digital audio broadcasting (DAB) and digital video broadcasting (DVB) systems. Also, blocks in the flowchart may be skipped or performed out of order and still produce desirable results. Accordingly, other embodiments are within the scope of the following claims. 

1. A method comprising: receiving a plurality of symbols; observing a plurality of data samples in adjacent symbols; and calculating, in a calculator, an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols, wherein calculating the estimate comprises calculating sum values corresponding to respective symbol indices, wherein each of the sum values is based on a summation of max values that correspond to respective data subcarrier indices, the max values being based on a maximum of (i) an absolute value of a real component of a base value and (ii) an absolute value of an imaginary component of the base value, wherein the base value is based on a multiplication of e^(−j2πnα) and at least one of the data samples, wherein n is a symbol index of the symbol indices, and wherein α is a cyclic prefix width ratio, wherein the symbols comprise Orthogonal Frequency Division Multiplexing (OFDM) symbols, and the plurality of data samples correspond to a plurality of data subcarriers in the OFDM symbols.
 2. The method of claim 1, wherein said calculating the estimate comprises solving the equation: ${\hat{l} = {\underset{n}{argmax}\left\{ {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}} \right\}}},$ where k is a subcarrier index, S2 is a set of indices for data subcarriers, and V is an observation vector.
 3. The method of claim 1, wherein each of the OFDM symbols further comprise a plurality of pilot samples corresponding to a plurality of pilot subcarriers in the OFDM symbols, and wherein said calculating comprises calculating the estimate of the integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data subcarriers and the pilot subcarriers between OFDM symbols.
 4. The method of claim 3, wherein said calculating the estimate comprises solving the equation: $\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}}} \right\}}$ where k is a subcarrier index, S1 is a set of indices for pilot subcarriers, S2 is a set of indices for data subcarriers, V is an observation vector, and A is a known sequence.
 5. The method of claim 1, further comprising: using the estimate of the integer portion of the carrier frequency offset to process a digital broadcast signal, wherein the digital broadcast signal comprises audio and/or video data.
 6. An apparatus comprising: a receiver to receive a plurality of symbols; a framer to observe a plurality of data samples in adjacent symbols; and a calculator to calculate an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols, wherein the calculator is operative to calculate the estimate by calculating sum values corresponding to respective symbol indices, wherein each of the sum values is based on a summation of max values that correspond to respective data subcarrier indices, the max values being based on a maximum of (i) an absolute value of a real component of a base value and (ii) an absolute value of an imaginary component of the base value, wherein the base value is based on a multiplication of e^(−j2πnα) and at least one of the data samples, wherein n is a symbol index of the symbol indices, and wherein α is a cyclic prefix width ratio, wherein the symbols comprise Orthogonal Frequency Division Multiplexing (OFDM) symbols, and the plurality of data samples correspond to a plurality of data subcarriers in the OFDM symbols.
 7. The apparatus of claim 6, wherein the calculator is operative to calculate the estimate by solving the equation: ${\hat{l} = {\underset{n}{argmax}\left\{ {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}} \right\}}},$ where k is a subcarrier index, S2 is a set of indices for data subcarriers, and V is an observation vector.
 8. The apparatus of claim 6, wherein each of the OFDM symbols further comprise a plurality of pilot samples corresponding to a plurality of pilot subcarriers in the OFDM symbols, and wherein the calculator is operative to calculate the estimate of the integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data subcarriers and the pilot subcarriers between OFDM symbols.
 9. The apparatus of claim 8, wherein the estimator is operative to calculate the estimate by solving the equation: $\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}}} \right\}}$ where k is a subcarrier index, S1 is a set of indices for pilot subcarriers, S2 is a set of indices for data subcarriers, V is an observation vector, and A is a known sequence.
 10. The apparatus of claim 6, wherein the symbols are included in a digital broadcast signal, wherein the digital broadcast signal comprises audio and/or video data.
 11. The apparatus of claim 6, further comprising: a downconverter communicatively coupled with the calculator to compensate for a cyclic shift in a received signal by using the estimate of the integer portion of the carrier frequency offset.
 12. A system comprising: an antenna to receive signals; and an estimator communicatively coupled with the antenna, the estimator including: a receiver to receive a plurality of symbols; a framer to observe a plurality of data samples in adjacent symbols; and a calculator to calculate an estimate of an integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data samples between symbols, wherein the calculator is operative to calculate the estimate by calculating sum values corresponding to respective symbol indices, wherein each of the sum values is based on a summation of max values that correspond to respective data subcarrier indices, the max values being based on a maximum of (i) an absolute value of a real component of a base value and (ii) an absolute value of an imaginary component of the base value, wherein the base value is based on a multiplication of e^(−j2πnα) and at least one of the data samples, wherein n is a symbol index of the symbol indices, and wherein α is a cyclic prefix width ratio, wherein the symbols comprise Orthogonal Frequency Division Multiplexing (OFDM) symbols, and the plurality of data samples correspond to a plurality of data subcarriers in the OFDM symbols.
 13. The system of claim 12, wherein the estimator is operative to calculate the estimate by solving the equation: ${\hat{l} = {\underset{n}{argmax}\left\{ {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}} \right\}}},$ where k is a subcarrier index, S2 is a set of indices for data subcarriers, and V is an observation vector.
 14. The system of claim 12, wherein each of the OFDM symbols further comprise a plurality of pilot samples corresponding to a plurality of pilot subcarriers in the OFDM symbols, and wherein the estimator is operative to calculate the estimate of the integer portion of a carrier frequency offset based on a cyclic shift and a phase shift of the data subcarriers and the pilot subcarriers between OFDM symbols.
 15. The system of claim 14, wherein the estimator is operative to calculate the estimate by solving the equation: $\hat{l} = {\underset{n}{argmax}\left\{ {{\sum\limits_{k \in S_{1}}^{\;}\;{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}{A^{*}\lbrack k\rbrack}} \right\}}} + {\sum\limits_{k \in S_{2}}^{\;}\;{\max\left\{ {{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}},{{\left\{ {{V\left\lbrack {k + n} \right\rbrack}{\mathbb{e}}^{{- {j2\pi}}\; n\;\alpha}} \right\}}}} \right\}}}} \right\}}$ where k is a subcarrier index, S1 is a set of indices for pilot subcarriers, S2 is a set of indices for data subcarriers, V is an observation vector, and A is a known sequence.
 16. The system of claim 12, wherein the symbols are included in a digital broadcast signal, wherein the digital broadcast signal comprises audio and/or video data.
 17. The system of claim 12, further comprising: a downconverter communicatively coupled with the estimator to compensate for a cyclic shift in a received signal by using the estimate of the integer portion of the carrier frequency offset. 